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author soto@cr.ie.u-ryukyu.ac.jp
date Tue, 08 Sep 2020 18:38:08 +0900
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0:b919985837a3 1:73127e0ab57c
1 module whileTestGears where
2
3 open import Function
4 open import Data.Nat
5 open import Data.Bool hiding ( _≟_ ; _≤?_ ; _≤_ ; _<_)
6 open import Data.Product
7 open import Level renaming ( suc to succ ; zero to Zero )
8 open import Relation.Nullary using (¬_; Dec; yes; no)
9 open import Relation.Binary.PropositionalEquality
10 open import Agda.Builtin.Unit
11
12 open import utilities
13 open _/\_
14
15 -- original codeGear (with non terminatinng )
16
17 record Env : Set (succ Zero) where
18 field
19 varn : ℕ
20 vari : ℕ
21 open Env
22
23 whileTest : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Env → t) → t
24 whileTest c10 next = next (record {varn = c10 ; vari = 0 } )
25
26 {-# TERMINATING #-}
27 whileLoop : {l : Level} {t : Set l} → Env → (Code : Env → t) → t
28 whileLoop env next with lt 0 (varn env)
29 whileLoop env next | false = next env
30 whileLoop env next | true =
31 whileLoop (record env {varn = (varn env) - 1 ; vari = (vari env) + 1}) next
32
33 test1 : Env
34 test1 = whileTest 10 (λ env → whileLoop env (λ env1 → env1 ))
35
36 proof1 : whileTest 10 (λ env → whileLoop env (λ e → (vari e) ≡ 10 ))
37 proof1 = refl
38
39 -- codeGear with pre-condtion and post-condition
40 --
41 -- ↓PostCondition
42 whileTest' : {l : Level} {t : Set l} → {c10 : ℕ } → (Code : (env : Env ) → ((vari env) ≡ 0) /\ ((varn env) ≡ c10) → t) → t
43 whileTest' {_} {_} {c10} next = next env proof2
44 where
45 env : Env
46 env = record {vari = 0 ; varn = c10 }
47 proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -- PostCondition
48 proof2 = record {pi1 = refl ; pi2 = refl}
49
50
51 open import Data.Empty
52 open import Data.Nat.Properties
53
54
55 {-# TERMINATING #-} -- ↓PreCondition(Invaliant)
56 whileLoop' : {l : Level} {t : Set l} → (env : Env ) → {c10 : ℕ } → ((varn env) + (vari env) ≡ c10) → (Code : Env → t) → t
57 whileLoop' env proof next with ( suc zero ≤? (varn env) )
58 whileLoop' env proof next | no p = next env
59 whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next
60 where
61 env1 = record env {varn = (varn env) - 1 ; vari = (vari env) + 1}
62 1<0 : 1 ≤ zero → ⊥
63 1<0 ()
64 proof3 : (suc zero ≤ (varn env)) → varn env1 + vari env1 ≡ c10
65 proof3 (s≤s lt) with varn env
66 proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p)
67 proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in
68 begin
69 n' + (vari env + 1)
70 ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩
71 n' + (1 + vari env )
72 ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩
73 (n' + 1) + vari env
74 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩
75 (suc n' ) + vari env
76 ≡⟨⟩
77 varn env + vari env
78 ≡⟨ proof ⟩
79 c10
80 ∎
81
82 -- Condition to Invariant
83 conversion1 : {l : Level} {t : Set l } → (env : Env ) → {c10 : ℕ } → ((vari env) ≡ 0) /\ ((varn env) ≡ c10)
84 → (Code : (env1 : Env ) → (varn env1 + vari env1 ≡ c10) → t) → t
85 conversion1 env {c10} p1 next = next env proof4
86 where
87 proof4 : varn env + vari env ≡ c10
88 proof4 = let open ≡-Reasoning in
89 begin
90 varn env + vari env
91 ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩
92 c10 + vari env
93 ≡⟨ cong ( λ n → c10 + n ) (pi1 p1 ) ⟩
94 c10 + 0
95 ≡⟨ +-sym {c10} {0} ⟩
96 c10
97 ∎
98
99 -- all proofs are connected
100 proofGears : {c10 : ℕ } → Set
101 proofGears {c10} = whileTest' {_} {_} {c10} (λ n p1 → conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 → ( vari n2 ≡ c10 ))))
102
103 --
104 -- codeGear with loop step and closed environment
105 --
106
107 open import Relation.Binary
108
109 record Envc : Set (succ Zero) where
110 field
111 c10 : ℕ
112 varn : ℕ
113 vari : ℕ
114 open Envc
115
116 whileTestP : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Envc → t) → t
117 whileTestP c10 next = next (record {varn = c10 ; vari = 0 ; c10 = c10 } )
118
119 whileLoopP : {l : Level} {t : Set l} → Envc → (next : Envc → t) → (exit : Envc → t) → t
120 whileLoopP env next exit with <-cmp 0 (varn env)
121 whileLoopP env next exit | tri≈ ¬a b ¬c = exit env
122 whileLoopP env next exit | tri< a ¬b ¬c =
123 next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 })
124
125 -- equivalent of whileLoopP but it looks like an induction on varn
126 whileLoopP' : {l : Level} {t : Set l} → (n : ℕ) → (env : Envc) → (n ≡ varn env) → (next : Envc → t) → (exit : Envc → t) → t
127 whileLoopP' zero env refl _ exit = exit env
128 whileLoopP' (suc n) env refl next _ = next (record {c10 = (c10 env) ; varn = varn env ; vari = suc (vari env) })
129
130 -- normal loop without termination
131 {-# TERMINATING #-}
132 loopP : {l : Level} {t : Set l} → Envc → (exit : Envc → t) → t
133 loopP env exit = whileLoopP env (λ env → loopP env exit ) exit
134
135 whileTestPCall : (c10 : ℕ ) → Envc
136 whileTestPCall c10 = whileTestP {_} {_} c10 (λ env → loopP env (λ env → env))
137
138 --
139 -- codeGears with states of condition
140 --
141 data whileTestState : Set where
142 s1 : whileTestState
143 s2 : whileTestState
144 sf : whileTestState
145
146 whileTestStateP : whileTestState → Envc → Set
147 whileTestStateP s1 env = (vari env ≡ 0) /\ (varn env ≡ c10 env)
148 whileTestStateP s2 env = (varn env + vari env ≡ c10 env)
149 whileTestStateP sf env = (vari env ≡ c10 env)
150
151 whileTestPwP : {l : Level} {t : Set l} → (c10 : ℕ) → ((env : Envc ) → whileTestStateP s1 env → t) → t
152 whileTestPwP c10 next = next env record { pi1 = refl ; pi2 = refl } where
153 env : Envc
154 env = whileTestP c10 ( λ env → env )
155
156 whileLoopPwP : {l : Level} {t : Set l} → (env : Envc ) → whileTestStateP s2 env
157 → (next : (env : Envc ) → whileTestStateP s2 env → t)
158 → (exit : (env : Envc ) → whileTestStateP sf env → t) → t
159 whileLoopPwP env s next exit with <-cmp 0 (varn env)
160 whileLoopPwP env s next exit | tri≈ ¬a b ¬c = exit env (lem (sym b) s)
161 where
162 lem : (varn env ≡ 0) → (varn env + vari env ≡ c10 env) → vari env ≡ c10 env
163 lem refl refl = refl
164 whileLoopPwP env s next exit | tri< a ¬b ¬c = next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) (proof5 a)
165 where
166 1<0 : 1 ≤ zero → ⊥
167 1<0 ()
168 proof5 : (suc zero ≤ (varn env)) → (varn env - 1) + (vari env + 1) ≡ c10 env
169 proof5 (s≤s lt) with varn env
170 proof5 (s≤s z≤n) | zero = ⊥-elim (1<0 a)
171 proof5 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in
172 begin
173 n' + (vari env + 1)
174 ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩
175 n' + (1 + vari env )
176 ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩
177 (n' + 1) + vari env
178 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩
179 (suc n' ) + vari env
180 ≡⟨⟩
181 varn env + vari env
182 ≡⟨ s ⟩
183 c10 env
184 ∎
185
186
187 whileLoopPwP' : {l : Level} {t : Set l} → (n : ℕ) → (env : Envc ) → (n ≡ varn env) → whileTestStateP s2 env
188 → (next : (env : Envc ) → (pred n ≡ varn env) → whileTestStateP s2 env → t)
189 → (exit : (env : Envc ) → whileTestStateP sf env → t) → t
190 whileLoopPwP' zero env refl refl next exit = exit env refl
191 whileLoopPwP' (suc n) env refl refl next exit = next (record env {varn = pred (varn env) ; vari = suc (vari env) }) refl (+-suc n (vari env))
192
193
194 {-# TERMINATING #-}
195 loopPwP : {l : Level} {t : Set l} → (env : Envc ) → whileTestStateP s2 env → (exit : (env : Envc ) → whileTestStateP sf env → t) → t
196 loopPwP env s exit = whileLoopPwP env s (λ env s → loopPwP env s exit ) exit
197
198
199 loopPwP' : {l : Level} {t : Set l} → (n : ℕ) → (env : Envc ) → (n ≡ varn env) → whileTestStateP s2 env → (exit : (env : Envc ) → whileTestStateP sf env → t) → t
200 loopPwP' zero env refl refl exit = exit env refl
201 loopPwP' (suc n) env refl refl exit = whileLoopPwP' (suc n) env refl refl (λ env x y → loopPwP' n env x y exit) exit
202
203
204 loopHelper : (n : ℕ) → (env : Envc ) → (eq : varn env ≡ n) → (seq : whileTestStateP s2 env) → loopPwP' n env (sym eq) seq λ env₁ x → (vari env₁ ≡ c10 env₁)
205 loopHelper zero env eq refl rewrite eq = refl
206 loopHelper (suc n) env eq refl rewrite eq = loopHelper n (record { c10 = suc (n + vari env) ; varn = n ; vari = suc (vari env) }) refl (+-suc n (vari env))
207
208
209 -- all codtions are correctly connected and required condtion is proved in the continuation
210 -- use required condition as t in (env → t) → t
211 --
212 whileTestPCallwP : (c : ℕ ) → Set
213 whileTestPCallwP c = whileTestPwP {_} {_} c ( λ env s → loopPwP env (conv env s) ( λ env s → vari env ≡ c10 env ) ) where
214 conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env
215 conv e record { pi1 = refl ; pi2 = refl } = +zero
216
217
218 whileTestPCallwP' : (c : ℕ ) → Set
219 whileTestPCallwP' c = whileTestPwP {_} {_} c (λ env s → loopPwP' (varn env) env refl (conv env s) ( λ env s → vari env ≡ c10 env ) ) where
220 conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env
221 conv e record { pi1 = refl ; pi2 = refl } = +zero
222
223 helperCallwP : (c : ℕ) → whileTestPCallwP' c
224 helperCallwP c = whileTestPwP {_} {_} c (λ env s → loopHelper c (record { c10 = c ; varn = c ; vari = zero }) refl +zero)
225
226 --
227 -- Using imply relation to make soundness explicit
228 -- termination is shown by induction on varn
229 --
230
231 data _implies_ (A B : Set ) : Set (succ Zero) where
232 proof : ( A → B ) → A implies B
233
234 whileTestPSem : (c : ℕ) → whileTestP c ( λ env → ⊤ implies (whileTestStateP s1 env) )
235 whileTestPSem c = proof ( λ _ → record { pi1 = refl ; pi2 = refl } )
236
237 whileTestPSemSound : (c : ℕ ) (output : Envc ) → output ≡ whileTestP c (λ e → e) → ⊤ implies ((vari output ≡ 0) /\ (varn output ≡ c))
238 whileTestPSemSound c output refl = whileTestPSem c
239
240
241 whileConvPSemSound : {l : Level} → (input : Envc) → (whileTestStateP s1 input ) implies (whileTestStateP s2 input)
242 whileConvPSemSound input = proof λ x → (conv input x) where
243 conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env
244 conv e record { pi1 = refl ; pi2 = refl } = +zero
245
246 loopPP : (n : ℕ) → (input : Envc ) → (n ≡ varn input) → Envc
247 loopPP zero input refl = input
248 loopPP (suc n) input refl =
249 loopPP n (record input { varn = pred (varn input) ; vari = suc (vari input)}) refl
250
251 whileLoopPSem : {l : Level} {t : Set l} → (input : Envc ) → whileTestStateP s2 input
252 → (next : (output : Envc ) → (whileTestStateP s2 input ) implies (whileTestStateP s2 output) → t)
253 → (exit : (output : Envc ) → (whileTestStateP s2 input ) implies (whileTestStateP sf output) → t) → t
254 whileLoopPSem env s next exit with varn env | s
255 ... | zero | _ = exit env (proof (λ z → z))
256 ... | (suc varn ) | refl = next ( record env { varn = varn ; vari = suc (vari env) } ) (proof λ x → +-suc varn (vari env) )
257
258 loopPPSem : (input output : Envc ) → output ≡ loopPP (varn input) input refl
259 → (whileTestStateP s2 input ) → (whileTestStateP s2 input ) implies (whileTestStateP sf output)
260 loopPPSem input output refl s2p = loopPPSemInduct (varn input) input refl refl s2p
261 where
262 lem : (n : ℕ) → (env : Envc) → n + suc (vari env) ≡ suc (n + vari env)
263 lem n env = +-suc (n) (vari env)
264 loopPPSemInduct : (n : ℕ) → (current : Envc) → (eq : n ≡ varn current) → (loopeq : output ≡ loopPP n current eq)
265 → (whileTestStateP s2 current ) → (whileTestStateP s2 current ) implies (whileTestStateP sf output)
266 loopPPSemInduct zero current refl loopeq refl rewrite loopeq = proof (λ x → refl)
267 loopPPSemInduct (suc n) current refl loopeq refl rewrite (sym (lem n current)) =
268 whileLoopPSem current refl
269 (λ output x → loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
270 (λ output x → loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
271
272 whileLoopPSemSound : {l : Level} → (input output : Envc )
273 → whileTestStateP s2 input
274 → output ≡ loopPP (varn input) input refl
275 → (whileTestStateP s2 input ) implies ( whileTestStateP sf output )
276 whileLoopPSemSound {l} input output pre eq = loopPPSem input output eq pre